Theorem rabeqsnd 29342

Description: Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)

Hypotheses

Ref	Expression
rabeqsnd.0	|- ( x = B -> ( ps <-> ch ) )
rabeqsnd.1	|- ( ph -> B e. A )
rabeqsnd.2	|- ( ph -> ch )
rabeqsnd.3	|- ( ( ( ph /\ x e. A ) /\ ps ) -> x = B )

Assertion

Ref	Expression
rabeqsnd	|- ( ph -> { x e. A | ps } = { B } )

Distinct variable groups:   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem rabeqsnd

Step	Hyp	Ref	Expression
1		rabeqsnd.3	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ ps ) -> x = B )
2	1	expl 648	. . . . 5 |- ( ph -> ( ( x e. A /\ ps ) -> x = B ) )
3	2	alrimiv 1855	. . . 4 |- ( ph -> A. x ( ( x e. A /\ ps ) -> x = B ) )
4		rabeqsnd.1	. . . . . . . 8 |- ( ph -> B e. A )
5		rabeqsnd.2	. . . . . . . 8 |- ( ph -> ch )
6	4, 5	jca 554	. . . . . . 7 |- ( ph -> ( B e. A /\ ch ) )
7	6	a1d 25	. . . . . 6 |- ( ph -> ( x = B -> ( B e. A /\ ch ) ) )
8	7	alrimiv 1855	. . . . 5 |- ( ph -> A. x ( x = B -> ( B e. A /\ ch ) ) )
9		eleq1 2689	. . . . . . . 8 |- ( x = B -> ( x e. A <-> B e. A ) )
10		rabeqsnd.0	. . . . . . . 8 |- ( x = B -> ( ps <-> ch ) )
11	9, 10	anbi12d 747	. . . . . . 7 |- ( x = B -> ( ( x e. A /\ ps ) <-> ( B e. A /\ ch ) ) )
12	11	pm5.74i 260	. . . . . 6 |- ( ( x = B -> ( x e. A /\ ps ) ) <-> ( x = B -> ( B e. A /\ ch ) ) )
13	12	albii 1747	. . . . 5 |- ( A. x ( x = B -> ( x e. A /\ ps ) ) <-> A. x ( x = B -> ( B e. A /\ ch ) ) )
14	8, 13	sylibr 224	. . . 4 |- ( ph -> A. x ( x = B -> ( x e. A /\ ps ) ) )
15	3, 14	jca 554	. . 3 |- ( ph -> ( A. x ( ( x e. A /\ ps ) -> x = B ) /\ A. x ( x = B -> ( x e. A /\ ps ) ) ) )
16		albiim 1816	. . 3 |- ( A. x ( ( x e. A /\ ps ) <-> x = B ) <-> ( A. x ( ( x e. A /\ ps ) -> x = B ) /\ A. x ( x = B -> ( x e. A /\ ps ) ) ) )
17	15, 16	sylibr 224	. 2 |- ( ph -> A. x ( ( x e. A /\ ps ) <-> x = B ) )
18		rabeqsn 4214	. 2 |- ( { x e. A | ps } = { B } <-> A. x ( ( x e. A /\ ps ) <-> x = B ) )
19	17, 18	sylibr 224	1 |- ( ph -> { x e. A | ps } = { B } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {crab 2916   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921  df-sn 4178

This theorem is referenced by:  repr0  30689